Name: _________________________________________
Expressing Geometric Properties with Equations:
Using proofs to prove simple geometric theorems algebraically
Exam
Multiple Choice:
What is the center of the circle whose equation is x 2 - 6x + y 2 +6y = -9
a. (3, -3)
b. (3, 3)
c. (-3, 3)
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d. (-3, -3)
What is the equation of the circle with center (4, -5) and radius 4?
a. (x + 4) 2 + (y – 5) 2 = 4
b. (x - 4) 2 + (y + 5) 2 =4
c. (x + 4) 2 + (y – 5) 2 = 16
d. (x - 4) 2 
+ (y + 5) 2 = 16
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Which of the following points along the directed segment from M to N partitions the
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segment in the ratio 3 to 1?
a.
b.
c.
d.
(1.25, 2.5)
(-2.5, 6.25)
(-2.25, 1.5)
(5.25, 1.5)
Short Answer
For the following, write the equation of the circle with the given center and radius.
Center: (1, 2); radius: 4
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Center: (-3, -5); radius: 7
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Center: (4, -6); radius: 2
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For the following, find the center and radius of the circle with the given equation,
and then graph the circle.
x 2 - 3x + y 2 = 16
Center: _________________
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Radius: _________________
x 2 - 4x + y 2 + 3y = 15
Center: _________________
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Radius: _________________
Explain how to determine the radius of a circle
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Building a Proof
Problem 1:
a) Draw a circle on the coordinate plane below that is centered at the origin,
and contains the point (0,4).
b) Determine the radius (r) of this circle:
r=__________________________
c) Use the radius and the coordinates of the center to write the equation of this
circle.
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d) Determine if the point (1, 15 ) satisfies the equation of the circle.
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e) Using your answers from (a-d), explain why (1, 15 ) does or does not lie on
the circle.
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f) Explain how you can prove that the point (2, 5 ) does not lie on the circle.
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Prove or disprove that the point ( 1, 4 ) lies on a circle that is centered at the origin
and contains the point (0,3).
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Find the coordinates of the point Q that lies along the directed line segment from
A (-3, -4) to B (5, 0) and partitions the segment in the ratio 2 to 3.
a) Convert the ratio (2 to 3) into a percent.
b) Find the slope for the segment AB .
c) Find the coordinates of Q
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Explain how you can check that the slope of AQ is equal to the slope of AB .
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Explain how you can use the distance formula to check that Q partitions AB in the
ratio 2 to 3.
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Find the coordinates of the point R that lies along the directed line segment
from
J (-2, 5) to K (2, -3) and partitions the segment in the ratio 4 to 1
a) Convert the ratio (4 to 1) into a percent.
b) Find the slope for the segment JK .
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c) Find the coordinates of R.
This coordinate plane is a map that shows a straight highway between two
towns. The highway planners want to build two new gas stations between the
towns so that the two gas stations divide the high way into three equal parts.
Find the coordinates of the points at which the gas stations should be built.
Prove or disprove that the quadrilateral determined by the points A (-2, 3), B
(5,3), C (3, -1), and D (-3, -1) is a parallelogram.
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Prove or disprove that the quadrilateral determined by the points Q (-2, 3), R
(3, 4), S (0, -2), and T (-4, -1) is a parallelogram.
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Explain how to prove or disprove that ABCD is not a square.
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Consider the points L (3,-4), M (1, -2), and N (5,2).
a. Find the coordinate points of P so that the quadrilateral determined by
L, M, N, and P is a parallelogram.
b. Is there more than one possibility? One that is a rectangle? Explain why
or why not.
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